@misc{StefanelliWachsmuthWachsmuth, author = {Stefanelli, Ulisse and Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {Optimal control of a rate-independent evolution equation via viscous regularization}, series = {Discrete and Continuous Dynamical Systems - Series S (DCDS-S)}, volume = {10}, journal = {Discrete and Continuous Dynamical Systems - Series S (DCDS-S)}, number = {6}, issn = {1937-1632}, doi = {10.3934/dcdss.2017076}, pages = {1467 -- 1485}, abstract = {We study the optimal control of a rate-independent system that is driven by a convex quadratic energy. Since the associated solution mapping is non-smooth, the analysis of such control problems is challenging. In order to derive optimality conditions, we study the regularization of the problem via a smoothing of the dissipation potential and via the addition of some viscosity. The resulting regularized optimal control problem is analyzed. By driving the regularization parameter to zero, we obtain a necessary optimality condition for the original, non-smooth problem.}, language = {en} } @misc{WachsmuthWachsmuth, author = {Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {Regularization error estimates and discrepancy principle for optimal control problems with inequality constraints}, series = {Control and Cybernetics}, volume = {40}, journal = {Control and Cybernetics}, number = {4}, issn = {0324-8569}, pages = {1125 -- 1158}, abstract = {In this article we study the regularization of optimization problems by Tikhonov regularization. The optimization problems are subject to pointwise inequality constraints in L²(Ω). We derive a-priori regularization error estimates if the regularization parameter as well as the noise level tend to zero. We rely on an assumption that is a combination of a source condition and of a structural assumption on the active sets. Moreover, we introduce a strategy to choose the regularization parameter in dependence of the noise level. We prove convergence of this parameter choice rule with optimal order.}, language = {en} } @misc{WachsmuthWachsmuth, author = {Wachsmuth, Gerd and Wachsmuth, Daniel}, title = {Convergence and Regularization Results for Optimal Control Problems with Sparsity Functional}, series = {ESAIM: Control, Optimisation and Calculus of Variations}, volume = {17}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, number = {3}, issn = {1292-8119}, doi = {10.1051/cocv/2010027}, pages = {858 -- 886}, abstract = {Optimization problems with convex but non-smooth cost functional subject to an elliptic partial differential equation are considered. The non-smoothness arises from a L1-norm in the objective functional. The problem is regularized to permit the use of the semi-smooth Newton method. Error estimates with respect to the regularization parameter are provided. Moreover, finite element approximations are studied. A-priori as well as a-posteriori error estimates are developed and confirmed by numerical experiments.}, language = {en} } @misc{GeigerWachsmuthWachsmuth, author = {Geiger, Tobias and Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {Optimal control of ODEs with state suprema}, series = {Mathematical Control and Related Fields}, volume = {11}, journal = {Mathematical Control and Related Fields}, number = {3}, issn = {2156-8472}, doi = {10.3934/mcrf.2021012}, pages = {555 -- 578}, abstract = {We consider the optimal control of a differential equation that involves the suprema of the state over some part of the history. In many applications, this non-smooth functional dependence is crucial for the successful modeling of real-world phenomena. We prove the existence of solutions and show that related problems may not possess optimal controls. Due to the non-smoothness in the state equation, we cannot obtain optimality conditions via standard theory. Therefore, we regularize the problem via a LogIntExp functional which generalizes the well-known LogSumExp. By passing to the limit with the regularization, we obtain an optimality system for the original problem. The theory is illustrated by some numerical experiments.}, language = {en} } @misc{WachsmuthWachsmuth, author = {Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {Necessary conditions for convergence rates of regularizations of optimal control problems}, series = {System Modeling and Optimization : 25th IFIP TC 7 Conference, CSMO 2011, Berlin, Germany, September 12-16, 2011, Revised Selected Papers}, journal = {System Modeling and Optimization : 25th IFIP TC 7 Conference, CSMO 2011, Berlin, Germany, September 12-16, 2011, Revised Selected Papers}, pages = {145 -- 154}, abstract = {We investigate the Tikhonov regularization of control constrained optimal control problems. We use a specialized source condition in combination with a condition on the active sets. In the case of high convergence rates, these conditions are necessary and sufficient.}, language = {en} } @misc{WachsmuthWachsmuth, author = {Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {How not to discretize the control}, series = {Special Issue:Joint 87th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM) and Deutsche Mathematiker-Vereinigung (DMV)}, volume = {16}, journal = {Special Issue:Joint 87th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM) and Deutsche Mathematiker-Vereinigung (DMV)}, number = {1}, address = {Braunschweig}, doi = {10.1002/pamm.201610385}, pages = {793 -- 794}, abstract = {In this short note, we address the discretization of optimal control problems with higher order polynomials. We develop a necessary and sufficient condition to ensure that weak limits of discrete feasible controls are feasible for the original problem. We show by means of a simple counterexample that a naive discretization by higher order polynomials can lead to non-feasible limits of sequences of discrete solutions.}, language = {en} } @misc{WachsmuthWachsmuth, author = {Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {A simple proof of the Baillon-Haddad theorem on open subsets of Hilbert spaces}, series = {Journal of Convex Analysis}, volume = {30}, journal = {Journal of Convex Analysis}, number = {4}, pages = {1319 -- 1328}, language = {en} } @misc{WachsmuthWachsmuth, author = {Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {A simple proof of the Baillon-Haddad theorem on open subsets of Hilbert spaces}, series = {arXiv}, journal = {arXiv}, pages = {1 -- 11}, language = {en} } @misc{WachsmuthWachsmuth, author = {Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {Second-order conditions for non-uniformly convex integrands: quadratic growth in L¹}, series = {Journal of Nonsmooth Analysis and Optimizaton}, volume = {3}, journal = {Journal of Nonsmooth Analysis and Optimizaton}, issn = {2700-7448}, doi = {10.46298/jnsao-2022-8733}, pages = {1 -- 36}, language = {en} } @misc{WachsmuthWachsmuth, author = {Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {Second-order conditions for non-uniformly convex integrands: quadratic growth in L^1}, series = {arXiv}, journal = {arXiv}, pages = {40}, abstract = {We study no-gap second-order optimality conditions for a non-uniformly convex and non-smooth integral functional. The integral functional is extended to the space of measures. The obtained second-order derivatives contain integrals on lower-dimensional manifolds. The proofs utilize the convex pre-conjugate, which is an integral functional on the space of continuous functions. Application to non-smooth optimal control problems are given.}, language = {en} }